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Basics of Research Part 13
Cheryl Bagley Thompson, PhD, RN
Descriptive Data Analysis
This 13th article of the Basics of Research series is first in a short series on statistical analysis. These articles
will discuss creating your statistical analysis plan, levels of measurement, descriptive statistics, probability theory, inferential statistics, and general considerations for interpretation of the results of a statistical analysis.
Statistical Analysis Plan
The most important part of any research project is the
planning process. This statement is as true for data analysis
as for any of the other steps in the research process. The
development of your statistical analysis plan should not be
delayed until after you have your data in hand. Rather, the
investigator should select the statistics to describe the sample and to analyze the data for each research question or
hypothesis before initiating the study. Most grant applications will require this information, but these decisions
should be made regardless of application for funding.
Investigators should plan to first describe their sample.
They should identify the important demographic characteristics of the sample, such as sex, age, and race. These variables will be the same for most studies. Other sample
characteristics, such as diagnosis, weight, height, Glasgow
Coma Scale, and so forth, also may be important to provide. Descriptive statistics will describe these variables.
Next, investigators should plan the analyses for each
research question/hypothesis. A table may be useful for this
activity. In the first column should be the research question/hypothesis; in the second, all relevant variables (and
timing information if needed); and in the third, the statistical test to be used. This process helps investigators ensure
that they are collecting all needed data, at the right time, to
answer their question. After all study data have been collected is not the time to discover that an important piece of
data has been missed.
Investigators uncomfortable with statistical analysis should
consult a statistician early in the planning phase. A statistician
will help them determine what statistical analyses are most
appropriate for answering the research questions/hypotheses,
taking into consideration the types of data to be collected.
Although statisticians may seem intimidating, investigators
should consider them an important member of the research
team and avail themselves of their expertise.
To help diminish the stress of a statistical consultation,
investigators should prepare a list of questions before the
meeting. In creating the list of questions, they need to start
with the research question.1 If the investigators have an
idea of what statistics to use, then the questions for the statistician are related to whether the proposed analyses are
appropriate and what other statistics should be considered.
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If the investigators have no idea of what statistics to use,
the first question should be what statistics are appropriate
for the research questions being addressed.
Investigators should take advantage of the meeting with
the statistician to find out why the analysis is appropriate
and to increase their knowledge of statistics. They need to
be able to defend their choice of statistics at presentations
and within publications.
Several advantages result from having a plan for data
analysis before starting the study. The most obvious is that
the investigators are not left wondering what to do with all
of the data they now have in their computer. A plan speeds
the process of data analysis. If a computer program will be
used, the commands for the analysis can even be written
before data collection is complete. In this case, as soon as all
of the data are entered, the investigators run the predetermined programs, and the analysis is ready for interpretation.
The second advantage of planning the statistical analysis
before the study is an increase in scientific integrity. The
investigators who have a plan ahead of time are less likely
to bend the analysis to suit their purpose. A plan also prevents the process of repeating analyses until something is
found that is statistically significant. A post hoc (after the
fact) approach to statistical analysis is inappropriate and
increases the chance of making a type I statistical error (see
a future issue of this series on hypothesis testing for a discussion of type I errors).2 If post hoc analyses are used, a
technique such as Bonferroni adjustment is needed to
decrease the chance of a type 1 error.2
Before beginning a study, the investigators should identify the computer, the data entry method,3 and the data
analysis software they will use for the study. They also
should spend time during the early phases of the project
becoming familiar with the software to be used. Data analysis will proceed more smoothly if the investigators do not
need to stop and ask for technical assistance.
Statistical Analysis
Overview
Statistical analysis can be a complex process. However, the
statistics required for studies most commonly done in critical
care transport research are fairly straightforward. Statistics are
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generally descriptive (describing what is) or inferential (determining the likelihood of a real difference being present in the
population). To select the most appropriate statistics, investigators need to know which type of question they are asking
and the level of measurement being used for the variables.
This article presents information on levels of measurement
and descriptive statistics, leaving probability theory and inferential statistics for a future issue.
Level of Measurement
Level of measurement refers to the amount of information contained within the data element and to some extent
the degree of detail present. Data elements are measured at
the nominal (categorical), ordinal, interval, or ratio (continuous) level. The term nominal level (or categorical) data
refers to data that can only be put into groups. For example, the demographic data elements of race and religion are
measured at the nominal level. This means that the values
consist of categories such as white, African American,
Native American, Asian, and other. With nominal level
data, no category is better than another, and the difference
between categories cannot be determined. For example,
neither Catholic nor Protestant is better than the other, and
whether Catholic is closer to Protestant or closer to Jewish
is not known. The only interpretation possible is that two
subjects are or are not the same on this variable.
A specific subset of nominal level data is dichotomous
data. Dichotomous data are nominal but have only two possible categories. A common example is mortality. The values
for mortality are either live or die. Other dichotomous variables are things that can be measured as yes or no, on or off,
or present or absent. Dichotomous variables possess characteristics beyond those of other nominal level data, but such
a discussion is beyond the scope of this article.
Ordinal level data are one step up from nominal data. As
the name implies, ordinal data have an inherent order. Data
values such as never, sometimes, often, and always have
order. An individual would interpret sometimes as being
more frequently than never and always as more frequently
than often. However, the difference in magnitude is not
known with ordinal level data. It cannot be said of ordinal
level data that always is twice as frequent as often or that the
distance between never and sometimes is the same as the distance between sometimes and often. The only interpretation
available is that of which is greater or which is lesser.
At the interval level of measurement, distances between
data elements can be determined. Temperature is the most
common variable measured at the interval level. With temperature, the difference between 40 and 50 degrees is the
same as the difference between 50 and 60 degrees.
However, interval level data have no true zero.
Consequently, multiplication is not allowed with interval
level data. This means that you cannot say that 10 degrees
is twice as hot as 5 degrees. Because there is no true zero, a
reference point does not exist. Zero degrees Fahrenheit or
Celsius are arbitrary numbers that do not relate to the
amount of temperature present. In contrast, temperature
measured in Kelvin has a true zero4 (absolute zero), where
March-April 2009
the absence of all energy serves as the reference point.
Unlike with Fahrenheit or Celsius, at 20 degrees Kelvin,
molecules are moving twice as fast as at 10 degrees Kelvin.
Data elements that have the characteristic of a true zero
are measured at the ratio (continuous) level of measurement. Common examples are height, weight, and heart
rate. Ratio level data are measured at the highest level of
measurement and contain the greatest amount of information. Ratio level data can be transformed by addition, subtraction, multiplication, or division without altering their
relative values. Ratio level data can be analyzed with the
widest range of statistical methods. Ratio level data are
often required for use with the most powerful statistics.2
Data elements can always be reduced in their level of measurement but can never be increased. For example, if income
is measured at the ratio level (in exact dollar amounts), the
data elements can be reduced to ordinal level data by the creation of categories (Table 1). However, data measured at the
ordinal level of measurement cannot be increased to the ratio
level of measurement. If we ask a subject what category her
income fits into, we can never determine from the raw data
her exact income level. Consequently, data should always be
collected at the highest level of measurement possible and
converted at the time of data analysis if a lower level of measurement is desired. This recommendation does not hold
when there is strong reason to believe that a subject will not
be truthful if she is required to provide exact data or when
the data cannot be expected to be accurately measured at the
higher level of measurement.
Descriptive Statistics
Descriptive statistics are numbers that summarize the
data with the purpose of describing what occurred in the
sample. In contrast, inferential statistics are numbers that
allow the investigator to determine whether there are differences between two or more samples and whether these
differences are likely to be present in the population of
interest. Descriptive statistics also can be used to compare
samples from one study with another. Descriptive statistics
also help researchers detect sample characteristics that may
influence their conclusions. For example, if a sample of air
medical personnel included 400 women and only 20 men,
the investigator would need to be careful about generalizing the findings to male air transport personnel.
Frequency distributions are often the first analyses to be
done on a data set. Frequency distributions are a valuable
Table 1. Yearly Income
Ratio Level Data
4,590
11,230
25,600
33,775
Ordinal Level Data
< 10,000
10,000-20,000
20,001-30,000
30,001-40,000
Data from the ratio level column can be reclassified as a value from the
ordinal level column but not vice versa.
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Table 2. Frequency Distribution
Gender
Number (%)
Male
154 (43.4)
Female
201 (56.6)
Figure 1. Frequency Distribution (Histogram)
Table 3. Bivariate Frequency Distribution
Professional Role
Gender
Paramedic
Nurse
Physician
Male
74
62
12
Female
53
142
5
Table 4. Mode for Interval or Ratio Level Data
Age
Frequency
18
5
23
1
31
1
40
3
42
4
43
2
45
1
46
2
54
1
65
1
Mode = 18
method for describing nominal or ordinal level data (discreet data). Because discreet data only characterize the
quantity within categories, a frequency distribution adequately describes nominal or ordinal level data. Frequency
distributions also can help detect data entry errors.
A frequency distribution consists of a description of the
number of subjects selecting each possible option and may
include the percentage of the sample that this number repre58
sents. For example, a frequency distribution for gender would
describe how many men and how many women were in the
sample. A frequency distribution can be shown using
numeric values or using graphical techniques (Table 2 and
Figure 1). Frequency distributions are often univariate (one
variable only) but may be bivariate (including two variables).
A bivariate frequency distribution is often presented as a table
with the name and values of one variable across the top and
the name and values of the second variable down the left side.
Table 3 is an example of a bivariate frequency distribution.
Multivariate frequency distributions describing more than
two variables at one time are possible but become more complex and are beyond the scope of this article.
Measures of central tendency are statistics that describe
where the middle of the sample lies. The lowest level measure
of central tendency is the mode. The mode is the value most
frequently occurring within the dataset. A mode can be used
with all levels of measurement and is the primary measure of
central tendency available for nominal level data. When
examining the number of patients with trauma, cardiac, or
other medical problems, the category having the most
patients represents the mode. If evaluating the ages of a sample, the age represented by the most subjects is the mode.
Although the mode provides helpful information for nominal or ordinal level data with only a few categories, the mode
may be of little value with interval or ratio level data. In
Table 4, age 18 is the most common age; however, the sample is generally much older than that. Overall the sample
consists mostly of individuals in their 40s, although not
many subjects have the exact same age in that small range.
The next measure of central tendency is the median, the
value that is in the exact middle of the sample. The median
is the point at which half of the subjects lie above this value
and half of the subjects lie below it. For example, the ages
of the nine subjects in sample 1 of Table 5 are arranged in
numeric order. The fifth age, 45, is in the exact middle of
the sample; consequently, age 45 is the median. The
median is a better measure of central tendency than mode
because it is not influenced by an accidental grouping of
values away from the true center of the data. However, the
median cannot be determined for nominal level data
because no order is present within the data.
The mean (or average) is the most common measure of
central tendency. The mean is calculated by adding up the
value for all subjects and dividing by the total number of
subjects (n). For sample 1 in Table 5, the mean is 38.3.
The mean is more sensitive to outliers and more influenced
by the distribution of the values than is the median. In Table
5 two data sets are demonstrated. Both have the same median
but have very different means. Consequently, different pieces
of information are available with the two measures, and one
or both may be relevant to the research at hand.
The use of means to describe a dataset should be limited to
interval and ratio level data. Nominal level data do not have a
true numeric value, so it is not possible to compute a mean.
Although ordinal data might be represented using numeric
values, the conceptual intervals between the values may not
be the same; therefore, the mean would be difficult to interAir Medical Journal 28:2
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Table 5. Median and Mean
Sample 1
18
20
21
36
45
46
52
53
54
Median
45
Mean
38.3
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Sample 2
34
36
38
42
45
55
60
58
62
45
47
Figure 2. Normal Distribution (Bell Curve)
Table 6. Range and Standard Deviation Example
Sample 1
Sample 2
19
33
31
42
42
48
44
51
55
53
61
54
62
56
69
60
76
62
Range
57
29
Mean
51
51
Standard Deviation
17.4
9.0
pret. For example, for never (0), occasionally (1), and always
(2), the conceptual difference between never and occasionally may not be the same as the difference between occasionally and always. Thus, the numbers 0, 1, and 2 do not
accurately represent the conceptual distance between values,
and the mean would be skewed accordingly.
Measures of central tendency provide information on where
the majority of data lie. However, these measures do not
inform the reader regarding the distribution of data across
possible values or their variability from one subject to the
next. One method for describing a collection of values is called
distribution. A normal distribution is typically described as
March-April 2009
being bell shaped, with a middle that is exactly in the center of
the distribution. In addition, the tails (sides) of the distribution are symmetric, having the exact same shape (Figure 2).
The presence of a normal distribution of data in the population is a common assumption for inferential statistics.
One measure of variability is the range. Range is the difference between the greatest value and the smallest value.
For the example in Table 6, the range in sample 1 is 57 and
in sample 2 is 29. The range informs the reader that one set
of data is more spread out (more variance) than the other.
The range, like the mean of a sample, is very sensitive to
outliers or measurements that are greatly different than the
rest of the sample. In Table 6, the range is wide, mostly
because of the influence of only two values, 19 and 76. The
rest of the values are clustered between 36 and 66, so a
range of 56 could be misleading.
A measure of variability that minimizes the effects of outliers is standard deviation. A standard deviation is a mathematical calculation of the variance of all the measurements in
a sample. The standard deviation can be viewed as the average
distance from the mean that each of the values lies. The mathematical equations for calculating the standard deviation can
be found in Burns and Grove2 and are easily performed by
basic statistical software programs or standard spreadsheets.
Pearson’s R is another common descriptive statistic. This
statistic describes the relationship between two variables.
Although Pearson’s R can be used descriptively, it is more
commonly used as an inferential statistic. This topic will be
covered in a later issue of the series.
Conclusion
In conclusion, I would like to stress that the best research
studies are initiated with a statistical plan already created.
This plan may or may not have been developed with the
assistance of a statistician. The first step of data analysis is
usually to describe the sample and then subgroups within
the sample. Frequency distribution, mean, median, mode,
range, and standard deviation are the most commonly used
statistics for accomplishing this task.
In the next issue in the series, the basics of probability
theory will be discussed. This information will be used as a
background to the discussion of inferential statistics.
References
1. Thompson CB, Panacek EA. Clinical research and critical care transport: How to get
started. Air Med J 2006;25:107-11.
2. Burns N, Grove SK. The practice of nursing research: Conduct, critique, and utilization. 5th ed. St. Louis: Elsevier Saunders; 2005.
3. Thompson CB, Panacek EA. Data management. Air Med J 2008;27:156-8.
4. U.S. Metric Association. Metric system temperature (Kelvin and degree Celsius).
Available at www.lamar.colostate.edu/~hillger/temps.htm. Accessed November 30,
2008.
Cheryl Bagley Thompson, PhD, RN, is an associate professor and
assistant dean of informatics and learning technologies at the
University of Nebraska Medical Center College of Nursing in
Omaha. She can be reached at cbthompson@unmc.edu.
1067-991X/$36.00
Copyright 2009 by Air Medical Journal Associates
doi:10.1016/j.amj.2008.12.001
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